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1) The article on covariant derivative in Wikipedia states that this is defined in terms of the Christoffel symbols as $\nabla_{e_i}\vec V=\left(\frac{\partial v^k}{\partial x^i}+v^j\Gamma_{ij}^k\right)e_k$. But the article on Christoffel symbols says these are defined by the equation $\nabla_i e_j=\Gamma_{ij}^k e_k$. To me, this seems circular.

2) I have seen people compute the Christoffel symbols associated with polar coordinates in the plane. They just compute things like $\frac{\partial \hat r}{\partial \theta}$ etc. How particular is this? Can these symbols always be computed by partial differentiation?

3) In his answer to question 270284 on MO, a user wrote the equations \begin{align*} &\nabla_y {\bf e}_x = -{\bf e}_y; \quad \nabla_y {\bf e}_y = {\bf e}_x\\ &\nabla_x {\bf e}_x = \nabla_x {\bf e}_y =0 \end{align*} I don't understand what exactly they mean. What is the difference between $\nabla_x$ and $\partial_x$ in this case? How are ${\bf e}_x$ and ${\bf e}_y$ related to the usual vectors $\partial_x$ and $\partial_y$?

4) The SO(3) generators $\{x\partial_y-y\partial_x,y\partial_z-z\partial_y,z\partial_x-x\partial_z\}$ are linearly independent almost everywhere, so suppose I want to use them as basis. What would be the Christoffel symbols in this case?

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  • $\begingroup$ 4): the Christoffel symbols of what? You need to have a metric or at least a connection to define Christoffel symbols $\endgroup$
    – Qfwfq
    May 30, 2017 at 17:11
  • $\begingroup$ @Qfwfq Concerning the assertion that I must have a connection to define Christoffel symbols, see questions 1) and 2) $\endgroup$
    – thedude
    May 30, 2017 at 18:54
  • $\begingroup$ @Qfwfq Regarding my question 4, could you please choose a connection, whichever you like, and then compute the Christoffel symbols, just so I can see how it works? $\endgroup$
    – thedude
    May 31, 2017 at 11:54

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I'll assume that the connection is the Levi-Civita connection of a Riemannian metric. If you aren't assuming that, you should clarify.

1) Given a set of local coordinates, there is an equivalence between a connection and its Christoffel symbols. If you have a connection, you can define the Christoffel symbols with respect to the coordinates. Conversely, if you have a set of coordinates and a set of Christoffel symbols, then you can use them to define Christoffel symbols. If you change coordinates, there is a formula for the Christoffel symbols with respect to the new coordinates in terms of the Christoffel symbols with respect to the old coordinates.

2) The Christoffel symbols of what connection?

3) Note that the definition of a connection $\nabla$ with respect to coordinates uses $\partial$. The latter is the partial derivative with respect to the coordinates and, if the Christoffel symbols do not all vanish, then $\nabla \ne \partial$.

4) To echo Qfwfq, when you ask "what are the Christoffel symbols?", your question makes sense only if you have a connection. Otherwise, the Christoffel symbols can be anything.

I strongly recommend that you find an introductory text to Riemannian geometry and learn more about metrics and connections from it. Wikipedia is not enough.

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  • $\begingroup$ Well, it still seems circular when you say I must have a connection in order to compute Christoffel symbols. How do I specify a connection without having such symbols in the first place? $\endgroup$
    – thedude
    May 30, 2017 at 19:28
  • $\begingroup$ I was studying a book, and the author says, we will compute the symbols of the polar coordinates of the plane. So he writes $\hat r=\cos\theta \hat i+\sin\theta\hat j$ and $\hat \theta=-r\sin\theta \hat i+r\cos\theta\hat j$ and he computes $\frac{\partial\hat r}{\partial\theta}=\frac{1}{r}\hat \theta$ to find $\Gamma_{r\theta}^\theta=\frac{1}{r}$. He never mention what is the connection, he just says let's compute the symbols of this coordinate system. $\endgroup$
    – thedude
    May 30, 2017 at 19:28
  • $\begingroup$ Can you tell me how $e_x$ and $e_y$ are related to the usual vectors $\partial_x$ and $\partial_y$ in point 3? $\endgroup$
    – thedude
    May 30, 2017 at 19:32
  • $\begingroup$ Which book are you studying? $\endgroup$
    – Deane
    May 30, 2017 at 21:48
  • $\begingroup$ I read bits and pieces from several, and I have seen the polar coordinates calculation in more than one. The one I have right now is Nakahara, Geometry, Topology and Physics (I am a physicist). But mathematical books on the subject are very abstract and lacking in terms of simple examples. Many of them just say oh, a connection is a linear map satisfying this and that property and just move on. $\endgroup$
    – thedude
    May 30, 2017 at 22:09

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